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Complete Two-loop Renormalization Group Equation of the Weinberg Operator

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Pith's one-line read The full two-loop renormalization group equation of the dimension-5 Weinberg operator in the Standard Model is derived, completing the two-loop RGE program of SMEFT up to dimension 5.

arxiv 2411.08011 v1 pith:WKFYQH5K submitted 2024-11-12 hep-ph hep-ex

Alejandro Ibarra , Nicholas Leister , Di Zhang This is my paper · ORCID
classification hep-phhep-ex
keywords operatortwo-loopweinbergcalculatecompleteequationgrouprenormalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Neutrinos have tiny masses that are not explained by the Standard Model. The simplest way to add them is a single quantum mechanical operator, the Weinberg operator, which violates lepton number and produces neutrino masses when the Higgs field gets a nonzero value. Its overall size and flavor structure change as the energy scale changes, and these changes are governed by a renormalization group equation, or RGE. This paper computes that RGE at two loops for the first time with only Standard Model particles in the loops. That means all quantum corrections are included to the next-to-leading order.

The calculation uses standard techniques: a background field method, dimensional regularization, and a trick in which a fake mass is introduced to separate infrared from ultraviolet divergences. The authors checked their result in several ways, for example by matching the one-loop pieces to earlier published results and by verifying a consistency relation between the first and second poles in the dimensional regulator.

The new equation's main physical consequence is that even if the lightest neutrino has zero mass at a very high energy scale, quantum effects generate a small mass at the electroweak scale, around 10^-13 eV for a cut-off of 10^14 GeV. The size of the generated mass depends on the Dirac and Majorana phases of the neutrino mixing matrix, and the associated Majorana phase is driven to a quasi-fixed point in the infrared. These effects are too small to be observed in planned experiments, but they give model builders complete, scheme-independent formulas.

Extended reading notes

Core claim

The paper's central assertion is that Eq. (21) is the complete two-loop contribution to the RGE of the Weinberg operator Wilson coefficient in the SM, 'thus completing the set of two-loop RGEs of the SM effective field theory up to dimension 5' (Abstract, Section 6). The load-bearing content is the analytic expression of Eq. (21), in particular the new gauge, Yukawa and Higgs-quartic terms that go beyond the rank-increasing (YlYl†)C5(YlYl†)^T term known from Ref. [25]. If the paper is correct, this is the first complete two-loop RGE of the dimension-5 Weinberg operator in the Standard Model.

Load-bearing premise

The completeness of the two-loop diagram enumeration for the C5 vertex: Section 4 states 'we calculate the divergent parts of all diagrams renormalizing C5 up to two-loops taking all external momenta to zero', but the paper only displays the topology list for the rank-increasing subset (Fig. 1); the exhaustive set for the full vertex is implicit in a FeynArts run (footnote 2). If the automatic generation missed a two-loop topology, Eq. (21) would not be the claimed complete RGE. This is a premise about toolchain and bookkeeping, distinct from the algebraic value of the result.

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Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central calculation has no fitted constants: the two-loop RGE, Eq. (21), depends only on SM gauge, Yukawa, and Higgs quartic couplings taken from the literature. The only numerical choices are the scenario scales Lambda = 10^14 GeV and Lambda_EW = 200 GeV used in Section 5 to illustrate the size of radiative corrections; these do not enter Eq. (21). The spurious regulator mass M for B, W and Higgs quantum fields is a technical device that cancels from all physical counterterms, so it does not appear in the ledger as an invented entity. No new particles, forces or conserved quantities are postulated.

free parameters (2)
  • Cut-off scale Lambda = 10^14 GeV (illustrative)
    Used in Section 5 (after Eq. 33 and in Fig. 2) to compute the generated lightest neutrino mass and phase correlations; it is a scenario input, not fitted to data, and does not enter the RGE itself.
  • Electroweak scale Lambda_EW = 200 GeV
    Lower boundary of the RG running in the numerical analysis (Fig. 2 caption); scenario input, not fitted and not part of Eq. (21).
assumptions (6)
  • standard math The beta function of a coupling is fully determined by the 1/epsilon pole of its renormalization constant in DR and MS-bar (Eq. 8).
    Used in Section 2 to convert the counterterm expansions Eqs. (5)-(8) into the RGEs; standard renormalization theory.
  • standard math The background field method with gauge fixing Eq. (2) yields a gauge-invariant background-field effective action and restricts field and coupling renormalization constants via Eq. (3).
    Section 2, following Refs. [38-40]; the foundation of the calculational setup.
  • domain assumption The infrared rearrangement with a common spurious mass M for the quantum B, W and Higgs fields separates infrared from ultraviolet divergences, so zero external momenta still give the correct UV counterterms.
    Section 4, approach of Ref. [50]; the cancellation of Delta-dependence in Eqs. (B5), (C3) and (C4) is the internal check.
  • domain assumption The two-loop RGEs of the SM parameters from Refs. [33-37] (Appendix D) are correct and complete.
    Used in the numerical analysis of Section 5 and in the completeness statement of Section 4; not rederived here.
  • domain assumption The automatic diagram generation with FeynArts enumerates all two-loop topologies contributing to the Weinberg operator vertex at vanishing external momenta.
    Section 4 and footnote 2; the exhaustive topology list is not printed, only the rank-increasing subset appears in Fig. 1.
  • domain assumption The tau-Yukawa dominance approximation, |xi12| << |xi11 - xi22| << 1, holds for the two-neutrino analytic solution and the 10^-13 eV estimates.
    Section 5.1, used to pass from Eq. (30) to Eq. (32) and to obtain the estimates in Eq. (33).

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Pith. "Pith review of Complete Two-loop Renormalization Group Equation of the Weinberg Operator." pith.science (2026). https://pith.science/paper/WKFYQH5K

@misc{pith2026241108011,
  author       = {Pith},
  title        = {Pith review of: Complete Two-loop Renormalization Group Equation of the Weinberg Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKFYQH5K}},
  note         = {Machine review of arXiv:2411.08011}
}
read the original abstract

We calculate the renormalization group equation (RGE) of the lepton-number-violating Weinberg operator with the particle content of the Standard Model (SM), thus completing the set of two-loop RGEs of the SM effective field theory up to dimension 5. We identify new diagrams that could increase the rank of the Wilson coefficient of the Weinberg operator, and we calculate the complete two-loop RGE for the neutrino mass eigenvalues and leptonic mixing matrix. We also briefly discuss some phenomenological implications of the RGEs.

Figures

Figures reproduced from arXiv: 2411.08011 by the authors.

Figure 1
Figure 1. Diagrams increasing the rank of neutrino mass matrix at the two-loop level. The last diagram results from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Contours of the lightest neutrino mass m1 (ΛEW) or m3 (ΛEW) with ΛEW = 200 GeV as a function of the two physical phases in the leptonic mixing matrix, the “Dirac phase” (δ) and the “Majorana phase” (σ for NMO and σ − ρ for IMO) at the scale Λ = 1014 GeV. which vanishes with the standard parametrization of the PMNS matrix from Ref. [60] (this is not necessarily the case for other parametrization [23, 61–63]). Therefo… view at source ↗
Figure 3
Figure 3. Contours of the Majorana phase ρ (ΛEW) associated with the lightest neutrino mass with ΛEW = 200 GeV in the physical phases [δ (Λ), σ (Λ)] or [δ (Λ), σ (Λ) − ρ (Λ)] plane with Λ = 1014 GeV for the NMO (the left panel) or IMO (the right panel). (a)NMO (b)IMO [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Evolution of the Majorana phase ρ from Λ = 1014 GeV to ΛEW = 200 GeV for different values of the lightest neutrino at the cut-off scale in the NMO (left panel) and IMO (right panel). Though the generated Majorana phase could be sizable, its effects are necessarily prop…

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum effects on neutrino parameters from a flavored gauge boson

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    A family-dependent U(1)_{Lμ-Lτ} gauge boson adds a one-loop term to the Weinberg-operator RGE that can raise the rank of the neutrino mass matrix, generating a lightest neutrino mass even if it starts at zero.

  2. Running of neutrino mass parameters in the Zee model

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    In the Zee model, one-loop EFT running—not direct matching—generates the neutrino mass matrix, and 1% variations in fitted high-scale couplings shift neutrino observables beyond next-generation experimental precision.

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    Including RGE effects in a global SMEFT fit improves bounds on poorly constrained four-quark operators but weakens limits on modified Z q qbar couplings by up to a factor of ten.

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    Introduction Neutrino oscillation experiments have established that the masses and mixing angles in the neutrino sector are qualitatively very different from those in the quark sector. A likely explanation for these differences is that neutrinos could be Majorana fermions rather than Dirac fermions. The lowest operator invariant under the Standard Model (...

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